How would I make a group vertices all the same distance from an Axis line?
A Cast modifier, set to 'Cylinder', with a Factor of 1, aimed at a vertex-group made of the selection you show, should do the job.

The Cylinder will be formed around the local Z axis of your object, or that of a control-object you allocate in the modifier.
It will be formed by pushing vertices outward toward the given radius. If the minimum is too large, the modifier may have to be applied, and the vertices scaled back to the desired final size.
The vertex Z-coordinates, in the cylinder's space, are not affected.

These tubes have been solidified for illustration.

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